What is the integral of θarctan(θ) ?

The integral of θarctan(θ) is 1/2 θ^2arctan(θ)-1/2 (-arctan(θ)+θ)+C

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integral of θarctan(θ)

\int θ arctan (θ) d θ

\frac{1}{2} θ^{2} arctan (θ) - \frac{1}{2} (- arctan (θ) + θ) + C

Solution steps

\int θ arctan (θ) d θ

Apply Integration By Parts

: \frac{1}{2} θ^{2} arctan (θ) - \int \frac{θ ^{2}}{2 ( θ ^{2} + 1 )} d θ

= \frac{1}{2} θ^{2} arctan (θ) - \int \frac{θ ^{2}}{2 ( θ ^{2} + 1 )} d θ

\int \frac{θ ^{2}}{2 ( θ ^{2} + 1 )} d θ = \frac{1}{2} (- arctan (θ) + θ)

= \frac{1}{2} θ^{2} arctan (θ) - \frac{1}{2} (- arctan (θ) + θ)

Add a constant to the solution

= \frac{1}{2} θ^{2} arctan (θ) - \frac{1}{2} (- arctan (θ) + θ) + C

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What is the integral of θarctan(θ) ?
The integral of θarctan(θ) is 1/2 θ^2arctan(θ)-1/2 (-arctan(θ)+θ)+C